I have a paper hosted on OSF.org that makes an attempt to resolve the Hubble tension and unify the dark sector amongst other challenges. Here is the abstract:
“This paper presents a first-principles derivation of the universal time dilation floor (𝑡𝐹) and the volumetric and 2D acceleration floors (𝑎𝐹 and 𝑎₀) from Planck power (𝑃𝐿). We demonstrate that these fundamental limits resolve several critical tensions in modern cosmology, including the Hubble tension, the primordial lithium problem, and the recent JWST impossible early galaxy observations. We identify 𝑎𝐹 and 𝑎0 as kinematic manifestations of the Planck power (𝑃𝐿 = 𝑐⁵/𝐺), which confers a property of metric stiffness on the spacetime manifold and represents the fundamental rate-limiting factor of cosmic energy transformation. Our framework suggests that dark energy and dark matter (the dark sector) are not independent entities, but rather different expressions of the same underlying physical process: the total energy transformation across all physical scales.”
The paper is hosted on OSF.org DOI https://doi.org/10.17605/OSF.IO/RCWJ3 I’ve attached a copy here as well. I’d be interested in any feedback, and apologies if I am in the wrong forum.
Andrew
The Emergent Time Framework Resolving the Dark Sector and Hubble Tension through Planck Power.pdf
Welcome to the forum @andrewcrussell You are posting on the right forum in terms of trying to develop a better cosmology. However your solution may not be popular with this group. Many of us consider the “dark sector” as just a hypothesis without physical justification. But not everyone shares this view.
My position is that an optical phenomenon causes various illusions which makes observations appear to need a “dark sector” to make sense. For example, the phenomenon produces a “cosmological redshift” which is in reality a light-electron interaction that shifts all spectra toward the red. The phenomenon also causes energy and momentum transfers between light and matter, which produce forces that appear to originate from the “dark sector”.
BTW, the ‘unify the dark sector’ means the paper shows that the effects of dark energy and dark matter emerge from the same fundamental underlying cause linked to cosmic expansion and the acceleration floor (a0 or aF depending on geometry). In LCDM they are treated as two separate phenomena – one is driven by a dark energy field and the other is looking for a dark matter particle.
I am in agreement that the dark sector hypotheses have some serious challenges that it is unlikely they will be able to solve. The paper takes a different tack, and instead of focusing on General Relativity’s usual ‘mass curves spacetime’ interpretation, focuses on the energy-time aspect and shows why dark matter and dark energy are not required to explain the Hubble tension, amongst other things.
MOND’s 1.2 x 10^-10 m/s^2 uses an interpolation function to model the radial acceleration relation (RAR) but does not provide an underlying explanation for a0. In the paper I posted, it is derived from first principles, and shows that this acceleration threshold scaled with the expansion of the cosmos (a(z) ~ c.H(z)). The higher acceleration in earlier epochs, and in regions with large clusters, is used to explain a range of problems that the dark matter/dark energy hypothesis cannot address.
There is another paper which applies similar principles to resolve the galaxy rotation anomaly.
On Sun, 12 Apr 2026 at 16:45, Louis Marmet @.***> wrote:
Here is the galaxy rotation anomaly paper – also hosted by OSF.org A Non-Dark Matter Solution to the Galaxy Radial Acceleration Relation V1.0.pdf
Focussing upon the math, I’ve scanned through your paper. And found some remarkable things. At first I calculated the CMB (Cosmic Microwave Background) temperature, according to the scheme suggested at page 3.
And the outcome is $\Theta \approx 54\mbox{ K}$ , instead of the expected $2.73\mbox{ K}$ . Did I make a mistake somewhere?
Hello Han, I think the discrepancy arises from the power flux vs. volumetric density distinction. In the ETF, Planck power is the total power at c. 3.6 x 10^52 watts driving the metric’s emergence. To find the temperature of the CMB, that power must be distributed over the 2D-surface of the expanding manifold rather than being treated as a local volumetric density which if I understand your response correctly is what you are showing. If you scale that power by the metric impedance, you have to divide the power by the surface area of the Hubble sphere 4 x pix RHubble^2 then factor in the 12pi geometric coefficient from the aF identity. You’ll find the energy density drops significantly, aligning the Stefan-Boltzmann result with the observed 2.73 K The excess energy you calculated is held in what in what I term “metric stiffness” in the paper i.e. it is the potential energy of the spacetime fabric itself, not the thermal energy of the photon background.
I get 2.9 K, which for a quick and dirty calculation is quite close.
My calculation is anyway wrong :-(That happens if one wants to do a quick post before going to sleep) An improved version would be aimed to look like this.
Resulting in temperatures (1,2,3) which are nowhere near the desired 2.73 K. Please elaborate what you did instead & exactly.
Han, Let’s treat the Planck power as the total source output driving the expansion of the manifold. To find the resulting background temperature, I distribute that total power across the total surface area of the Hubble horizon. In this framework, because of the 3 degrees of freedom in the manifold’s geometry, the total effective surface area is not the standard 4 * pi * R_Hubble^2, but is 12 * pi * R_Hubble^2. This accounts for the manifold’s 3 degrees of spatial freedom acting in unison. We can treat this as a surface power-loading problem like this
I think this question will make a useful basis for a write up to explain the thinking behind the calculation.
On page 8 of your paper you write, for the deflection angle $\alpha$ of a light ray past the sun, according to General Relativity:
$$ \alpha_{GR} = \frac{4GM}{bc^2} $$
Then you write, for the same deflection angle $\alpha$ of a light ray past the sun, according to the Emergent Time Framework:
$$ \alpha_{ETF} = \frac{4(c^5/P_L)M}{bc^2} = \frac{4Mc^3}{bP_L} $$
where the Planck power is defined already on page 2 as:
$$ P_L = c^5/G \quad \Longrightarrow \quad c^5/P_L = G $$
Thus I would have immediately concluded that
$$ \alpha_{ETF} = \frac{4(c^5/P_L)M}{bc^2} = \frac{4GM}{bc^2} = \alpha_{GR} $$
instead of making that longwinded numerical turn around trip on the pages 8 and 9.
That’s a fair comment - it does come across a bit pedantic. . The thrust of what I was trying to convey is that in GR, you don’t have to ‘fixate’ on mass curving spacetime when the calculation clearly shows energy also curves spacetime, and as I wrote in the initial paper in December (it’s up on the OSF and rxiVerse as well), you cannot have energy transformation or exchange without time, so it is ok to interpret GR in my view through the lens of energy and time and not just mass and spacetime curvature. LCDM fixes everything by adding non-baryonic mass and it’s led to a lot of problems (warm dark matter, fuzzy dark matter, halo-baryon conspiracy, AGN ‘clearing’ etc.)—everyone requires more free parameters. In the ETF, I try to get rid of the free parameters altogether, so it is a zero free parameter approach. In the Galaxy paper I point out that there are billions of galaxies that all have very similar structure and, as far as we know, all show the same a(z) acceleration floor characteristics—this is hardly a situation coming from lots of free parameters or a happy accident.
For my last comment I need the assistance of @RedshiftDrift as well. On the pages 5 and 12 of your paper you write for the MOND acceleration:
$$ a_0 = 2\pi a_F = \frac{cH_0}{6} \approx 1.133 \times 10^{−10}\mbox{ }m/s^2 $$
But on page 13 of the same paper you write:
$$ a_0 = 6 a_F = \frac{cH}{2\pi} $$
The latter formula would be in concordance with a result obtained by @RedshiftDrift , but I can’t find the reference so quickly. It’s for sure that $2\pi\ne 6$ , but I didn’t carefully read the accompanying proze, so it may be that I missed something, such as $H\ne H_0$ . Would you be so kind to elaborate on this “paradox”?
That’s a good point you raised, and it has to do with the relationship between a 2D constrained planar object (a galaxy) and a 3D manifold (the vacuum). On pages 5 and 12, we are looking at local galaxy dynamics. In these systems, the baryonic matter is constrained in a 2D orbital plane (the galaxy disk), and the acceleration floor a0 expresses circumferentially. Here, we use 2 * pi because we are measuring the metric’s influence over a flat, circular path.
On page 13, the focus shifts to the Hubble expansion. Here, we are no longer looking at a single 2D orbital slice, but at the manifold expanding in all three spatial dimensions. As the expansion acts along the 6 primary axes (+x, -x, +y, -y, +z, -z), the geometric coefficient becomes 6.
The relationship 2 * pi ~ 6.28 is why these two formulas appear so similar. In the local, flat environment of a galaxy, the circular geometry dominates, but in the 3D manifold, the vector components are free to operate on 6 degrees of freedom. The formula a0 = c * H / (2 * pi) is the 2D constrained expression of the law, and a0 = 6 * aF is the 3D volumetric expression. The difference between them (~ 4.5%) is the geometric curvature correction required when moving from a 2D constrained slice to the 3D manifold expansion.
If/when I do a revision, I will make this clearer with a footnote.
@HanDeBruijn the paper where I calculate $a_0 \approx cH/(2\pi)$ was rejected last year and I did not fight it. But the equation is also found empirically by Milgrom, McGaugh, etc… If I remember correctly, they don’t have the experimental accuracy to distinguish between $a_0 \approx cH/(2\pi)$ and $a_0 \approx cH/6$.
In this reply @andrewcrussell writes: >In this framework, because of the 3 degrees of freedom in the manifold’s geometry, the total effective surface area is not the standard $4\pi R_{Hubble}^2$ , but is $12\pi R_{Hubble}^2$ . This accounts for the manifold’s 3 degrees of spatial freedom acting in unison.
But in this reply the same author writes: >Here, we are no longer looking at a single 2D orbital slice, but at the manifold expanding in all three spatial dimensions. As the expansion acts along the 6 primary axes (+x, -x, +y, -y, +z, -z), the geometric coefficient becomes 6.
I don’t want to sound pedantic again, but in both cases there are three degrees of spatial freedom, so I would expect (3 XOR 6) in both cases. And - by the way - a (multiplication XOR division) in both cases.
I’ve put another paper up on the OSF that explores the 6 vs 2.pi topic https://doi.org/10.17605/OSF.IO/SFYNX
I’ve also put the papers up on my website https://emergent-time-framework.org/ where there’s a blog and some background about the ETF.
:)
I don’t quite know what you find confusing about it. The 2π vs 6 factor appears to be a source of confusion for people who have recognised a link between c.Ho/2.Pi or c.Ho/6. The paper explains how these two factors are linked – one describing the manifold 3D > 2D nucleation factor and the other the energy partitioning. :)
Follow this link to the Pioneer Anomaly discussion for a third possibility.How does the Pioneer anomaly lead to a solution for the RAR or the Tully-Fisher relationship from first principles?
@andrewcrussell , @RedshiftDrift .
Sorry: right answer in the wrong place !!
Quite unexpectedly, theory as published in Variable Mass Theory and Observable Universe [3] provides us with another approach to the 6 vs 2.pi topic. But the discovery is too immature to publish it at another medium than ACG Discussions. We are using the formulas (3), (11) and (12) in [3]:
$$ R_O = \frac{2GM_O}{c^2} \quad ; \quad R_O = W\frac{c}{H} \quad ; \quad M_O = \frac{c^3W}{2GH} $$
where $R_O=$ radius of Observable universe, $G=$ gravitational constant, $M_O=$ mass of Observable universe, $c=$ speed of light in vacuum, $H=$ Hubble constant = 71 km/s/Mpc, $W=$ 3.383634283 (typical for VM). The above formulas are mutually consistent, because
$$ M_O = \frac{c^2R_O}{2G} = \frac{c^2}{2G}.W\frac{c}{H} = \frac{c^3W}{2GH} $$
Next we calculate the acceleration $a_0$ at the horizon of any (!!) Observable universe.
$$ a_0 = \frac{GM_O}{R_O^2} = G\frac{c^3W}{2GH}.\left(\frac{H}{c.W}\right)^2 = \frac{Hc}{2W} $$
If that outcome is relevant for the topic at hand, then we have the following possibilities.
$$ a_0 = Hc \times \begin{cases} 1/6 & \approx 0.16667 \ 1/(2\pi) & \approx 0.15915 \ 1/(2W) & \approx 0.14777 \end{cases} $$
I said “IF”.